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Chapter 9: Problem 54

Evaluate. $$ 10^{2} $$

Short Answer

Expert verified
100.

Step by step solution

01

Identify the Base and Exponent

The given expression is \(10^2\). Here, 10 is the base and 2 is the exponent.
02

Apply the Exponentiation

Exponentiation means multiplying the base by itself as many times as the exponent indicates. In this case, you need to multiply 10 by itself 2 times.
03

Perform the Multiplication

Calculate \(10 \times 10\). The result is 100.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base and Exponent
When we talk about exponents, it's essential to understand the roles of the base and the exponent. In the expression \(10^2\), the number 10 is called the base. This is the number that will be repeatedly multiplied.
The number 2, placed as a superscript, is called the exponent. It tells us how many times to multiply the base by itself.
In our example, \(10^2\) means we will have 10 multiplied by itself 2 times. This might seem straightforward, but grasping these definitions is crucial for solving more complex problems using exponents.
Multiplication
At the heart of exponentiation is multiplication. Once you identify the base and the exponent, the next step is to perform the multiplication.
For \(10^2\), you multiply the base (10) by itself 2 times:
  • First, you start with the base number: 10.
  • Then, you multiply 10 by itself once: \(10 \times 10 = 100\).
Notice how we only had to perform one multiplication because the exponent was 2. As you handle higher exponents, you'll repeat the multiplication process more times.
Remember, multiplication is simply repeated addition. So \(10 \times 10\) means adding 10 to itself: \(10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10\), which indeed equals 100.
Powers of Ten
Exponentiation becomes especially interesting with powers of ten. Understanding powers of ten can help simplify many mathematical concepts.
When the base is 10 and the exponent is a positive integer, the result is always a 1 followed by a number of zeroes equal to the exponent.
For example:
  • \(10^1 = 10\) (1 zero)
  • \(10^2 = 100\) (2 zeroes)
  • \(10^3 = 1000\) (3 zeroes)
This pattern makes multiplying and dividing by powers of ten very straightforward. Because it's consistent, you can easily extend this knowledge to work with higher powers.
This rule is only for base 10 and positive exponents. It helps students quickly grasp the result without performing multiple multiplications.

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